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We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any &#8220;heat bath&#8222; (<inline-formula><math display="inline"><semantics><mrow><mi>h</mi> <mi>b</mi></mrow></semantics></math></inline-formula>), evolving through the Liouville equation for the non-equilibrium classical distribution <inline-formula> <math display="inline"> <semantics> <msub> <mi>W</mi> <mi>c</mi></msub></semantics></math></inline-formula>, with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann&#8217;s Gaussian classical equilibrium distribution <inline-formula> <math display="inline"> <semantics> <msub> <mi>W</mi> <mrow> <mi>c</mi> <mo>,</mo> <mi>e</mi> <mi>q</mi></mrow></msub></semantics></math></inline-formula>, as weight function to generate orthogonal polynomials (<inline-formula><math display="inline"><semantics><msub><mi>H</mi> <mi>n</mi></msub></semantics></math></inline-formula>&#8217;s) in momenta. The moments of <inline-formula> <math display="inline"> <semantics> <msub> <mi>W</mi> <mi>c</mi></msub></semantics></math></inline-formula>, implied by the <inline-formula> <math display="inline"> <semantics> <msub> <mi>H</mi> <mi>n</mi></msub></semantics></math></inline-formula>&#8217;s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a <inline-formula> <math display="inline"> <semantics> <mrow> <mi>h</mi> <mi>b</mi> </mrow> </semantics> </math> </inline-formula> are studied classically and quantum-mechanically (by using Wigner functions <i>W</i>). Difficulties related to the non-positivity of <i>W</i> are bypassed. Equilibrium Wigner functions <inline-formula> <math display="inline"> <semantics> <msub> <mi>W</mi> <mrow> <mi>e</mi> <mi>q</mi> </mrow> </msub> </semantics> </math> </inline-formula> generate orthogonal polynomials, which yield non-equilibrium moments of <i>W</i> and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new.